Osmania University College for Women, Koti, Hyderabad. B ... · OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABAD B.Sc II YEAR SYLLABUS, MATHEMATICS 2017-18 REAL ANALYSIS, SEMESTER-III,

Osmania University College for Women, Koti, Hyderabad.B.Sc (Mathematics)

Under CBCS Pattern

The Course of study and scheme of examination:

Semester Paper Title of the Paper hrs/week

Credits Internal marks

External marks

Total marks

Sem I Paper I Differential Calculus 4 Th3 Pr

41

10+7+5 3518

75

Sem II Paper II.Differential Equations

4 Th3 Pr

41

10+7+5 3518

75

Sem III Paper III Real analysis.4 Th3 Pr

41

10+7+5 3518

75

Sem IV Paper IV Algebra 4 Th3 Pr

41

10+7+5 3518

75

Sem V Paper V Linear Algebra 3 Th3 Pr

31

10+7+5 3518

75

Sem V Paper VI (A)

Paper VI (B)

Solid Geometry

Integral Calculus

3 Th3 Pr

3 Th3 Pr

31

31

10+7+5

10+7+5

3518

3518

75

75

Sem VI Paper VII Numerical Analysis 3 Th3 Pr

31

10+7+5 3518

75

Sem VI Paper VIII (A)

Paper VIII(B)

Complex Analysis

Vector Calculus

3 Th3 Pr

3 Th3 Pr

31

31

10+7+5

10+7+5

3518

3518

75

75

Osmania University College for Women, Koti, Hyderabad.B.Sc (Mathematics)

Under CBCS Pattern (2017-2018)

The Course of study and scheme of examination:

Semester Paper Title of the Paper hrs/week

Credits Internal marks

External marks

Total marks

Sem I Paper I Differential Calculus 4 Th3 Pr

41

10+7+5 3518

75

Sem II Paper II.Differential Equations

4 Th3 Pr

41

10+7+5 3518

75

Sem III Paper III Real analysis.

4 Th3 Pr

41

10+7+5 3518

75

Sem IV Paper IV Algebra 4 Th3 Pr

41

10+7+5 3518

75

Sem V Paper V Linear Algebra and Multiple Integrals

3 Th3 Pr

31

10+7+5 3518

75

Sem V Paper VI (A) Numerical Analysis 3 Th3 Pr

31

10+7+5 3518

75

Sem VI Paper VII Linear Algebra and Vector Calculus

3 Th3 Pr

31

10+7+5 3518

75

Sem VI Paper VIII (A) Integral Transforms 3 Th3 Pr

31

10+7+5 3518

75

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc I YEAR SYLLABUS MATHEMATICS 2017-18

DIFFERENTIAL CALCULUS, SEMESTER-I, PAPER-IUnit- I:Successive differentiation – Calculation of nth derivative of some standard results – Leibnitz theorem – nth derivative of product of two functions. Expansion of functions – Taylor, Maclaurin’s theorems.Concepts of limit, continuity, derivative at a point – Mean value theorems – Rolle’s, Lagrange’s and Cauchy mean value theorems( with Geometrical meaning and applications).Unit – II:Indeterminate forms – , , ∞ − ∞, 0 , 0 , 1 , ∞ -- L’Hospital’s Rule, Definition of

Curvature- Length of arc-- derivative of arc – Radius of curvature ρ (Cartesian, parametric and polar forms). Newtonian method. Generalized Newtonian formula .Centre of curvature –Evolutes and Involutes.

Unit – III:Partial differentiation-Functions of two variables- Partial derivative at point – Partial derivatives of higher orders, Taylor’s theorem for two variables. Homogeneous functions –Euler’s theorem on homogeneous functions, Total derivative – Differentiation of composite and implicit functions.

Unit – IV:Maxima and Minima of functions of two variables – conditions for the existence of maxima or minima, stationary and extreme points, Lagrange’s condition for maxima or minima of two variables – method of undetermined multipliers. Asymptotes (Cartesian and polar forms), Envelopes – determination of envelopes when parameters are connected by a relations (Cartesian form & Polar form), Envelope of normals (Evolutes).

Text: Shanti Narayan and Mittal, Differential Calculus , S.Chand & Co.References: Manicavachagam Pillay – Differential CalculusWilliam Anthony Granville, Percey F Smith and William Raymond Longley – Elements of the differential and integral calculusJoseph Edwards – Differential calculus for beginnersSmith and Minton – CalculusElis Pine – How to Enjoy CalculusHari Kishan – Differential Calculus

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc I YEAR SYLLABUS .MATHEMATICS 2017-18

DIFFERENTIAL EQUATIONS, SEMESTER-II, PAPER-II.

Unit – I:Differential Equations of first order and first degree: Exact differential equations –Integrating Factors – Change in variables – Total Differential Equations – Simultaneous Total

Differential Equations – Equations of the form = = Differential Equations first order but not of first degree: Equations Solvable for y – Equations Solvable for x – Equations that do not contain x (or y) – Clairaut’s equation.

Unit – II:Higher order linear differential equations: Solution of homogeneous linear differentialequations with constant coefficients – Solution of non-homogeneous differential equations P(D)y = Q(x) with constant coefficients by means of polynomial operators when Q(x) =bx , be , e V, b cos(ax) , bsin(ax)Unit – III:Method of undetermined coefficients – Method of variation of parameters – Lineardifferential equations with non constant coefficients – The Cauchy – Euler Equation, Legendre Equation, System of simulataneous linear differential equations, Degenerate case.

Unit – IV:Partial Differential equations- Formation and solution- Equations easily integrable –Linear equations of first order – Non linear equations of first order – Charpit’s method –Homogeneous linear partial differential equations with constant coefficients,Non homogeneous linear partial differential equations – Separation of variables

Text: Zafar Ahsan – Differential Equations and Their Applications

References: Frank Ayres Jr. – Theory and Problems of Differential EquationsFord, L.R. – Differential Equations.Daniel Murray – Differential EquationsS. Balachandra Rao – Differential Equations with Applications and ProgramsStuart P Hastings, J Bryce McLead – Classical Methods in Ordinary Differential

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc II YEAR SYLLABUS, MATHEMATICS 2017-18

REAL ANALYSIS, SEMESTER-III, PAPER-III Objective: The course is aimed at exposing the students to the foundations of analysis which

will be useful in understanding various physical phenomena.Outcome: After the completion of the course students will be in a position to appreciate beauty and applicability of the course.

Unit- ISequences: Limits of Sequences- A Discussion about Proofs-Limit Theorems for

Sequences-Monotone Sequences and Cauchy Sequences.

Unit- IISubsequences-Lim sup's and Lim inf's-Series-Alternating Series and Integral Tests.

Unit- IIISequences and Series of Functions: Power Series-Uniform Convergence-More on

Uniform Convergence-Differentiation and Integration of Power Series (Theorems in this section without Proofs).

Unit- IVIntegration: The Riemann Integral - Properties of Riemann Integral-Fundamental

Theorem of Calculus.

Text:Kenneth A Ross, Elementary Analysis-The Theory of Calculus

References:William F. Trench, Introduction to Real AnalysisLee Larson , Introduction to Real Analysis IShanti Narayan and Mittal, Mathematical AnalysisBrian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner; Elementary Real analysisSudhir R., Ghorpade, Balmohan V., Limaye; A Course in Calculus and Real Analysis

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc II YEAR SYLLABUS, MATHEMATICS 2017-18

ALGEBRA, SEMESTER-IV, PAPER-IV

Objective: The course is aimed at exposing the students to learn some basic algebraic structures like groups, rings etc. Outcome: On successful completion of the course students willbe able to recognize algebraic structures that arise in matrix algebra, linear algebra and will be able to apply the skills learnt in understanding various such subjects.

Unit- IGroups: Definition and Examples of Groups- Elementary Properties of Groups-Finite Groups;Subgroups -Terminology and Notation -Subgroup Tests - Examples of Subgroups Cyclic Groups:Properties of Cyclic Groups - Classification of Subgroups Cyclic Groups-Permutation Groups:Definition and Notation -Cycle Notation-Properties of Permutations -A Check Digit Scheme Based on D5.

Unit- IIIsomorphisms ; Motivation- Definition and Examples -Cayley's theorem Properties of Isomorphisms- Automorphisms-Cosets and Lagrange's Theorem Properties of Cosets 138 -Lagrange's Theorem and Consequences-An Application of Cosets to Permutation Groups -The Rotation Group of aCube and a Soccer Ball -Normal Subgroups and Factor Groups ; Normal Subgroups-Factor Groups-Applications of Factor Groups -Group Homomorphisms - Definition and Examples -Properties of Homomorphisms -The First Isomorphism Theorem.Unit- IIIIntroduction to Rings: Motivation and Definition -Examples of Rings -Properties of Rings -Subrings-Integral Domains : Definition and Examples {Characteristics of a Ring -Ideals and Factor Rings, Ideals -Factor Rings -Prime Ideals and Maximal Ideals.Unit- IVRing Homomorphisms: Definition and Examples-Properties of Ring- Homomorphisms -The Field of Quotients Polynomial Rings: Notation and Terminology.

Text:Joseph A Gallian, Contemporary Abstract algebra (9th edition)

References:Bhattacharya, P.B Jain, S.K.; and Nagpaul, S.R,Basic Abstract AlgebraFraleigh, J.B, A First Course in Abstract Algebra.Herstein, I.N, Topics in AlgebraRobert B. Ash, Basic Abstract AlgebraI Martin Isaacs, Finite Group Theory

Joseph J Rotman, Advanced Modern Algebra

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc III YEAR MATHEMATICS SYLLABUS 2017-18

LINEAR ALGEBRA & MULTIPLE INTEGRALS

SEMESTER-V PAPER-VObjectives:This course aims to study the Algebraic structures of Vector Spaces and to understand several important concepts in linear algebra, linear independence of vectors; subspaces, bases, and dimension of vector spaces; linear transformations and establishes different dimension theorems.This course also gives the concepts of integration in ℝ and ℝ and the technique to compute the length of curves and area of surfaces.Linear Algebra UNIT-I: Vector spaces, General properties of vector spaces, Vector subspaces, Algebra of subspaces, linear combination of vectors. Linear span, linear sum of two subspaces, Linear independence and dependence of vectors, Basis of vector space.

UNIT-II: Finite dimensional vector spaces, Dimension of a vector space, Dimension of a subspace. Linear transformations, linear operators, Range and null space of linear transformation, Rank and nullity of linear transformations, Linear transformations as vectors, Product of linear transformations, Invertible linear transformation. Matrix of linear transformations.

Prescribed text book:Linear Algebra by J.N.Sharma and A.R.Vasista, Krishna Prakasham Mandir, Meerut-250002Reference Books: 1. Linear Algebra by Kenneth Hoffman and Ray Kunze, Pearson Education (low priced edition), New Delhi2. Linear Algebra by Stephen H. Friedberg et al Prentice Hall of India Pvt. Ltd. 4th edition 2007

Multiple integralsUNIT-III: Multiple integrals, Introduction, the concept of a plane, Curve, line integral- Sufficient condition for the existence of the integral. The area of a subset of 2R , Calculation of double integrals, Jordan curve, Area, Change of the order of integration.

UNIT-IV:Double integral as a limit, Change of variable in a double integration. Lengths of Curves, surface areas, Integral expression for the length of a curve, surfaces, surface areas.

Prescribed text Book: Vector Analysis by Murray. R.Spiegel, Schaum Series Publishing Company.Reference Books: 1.Text book of vector Analysis by Shanti Narayana and P. K. Mittal,S. Chand & Company Ltd, New Delhi.2. Mathematical Analysis by S.C. Mallik and Savitha Arora, Wiley Eastern Ltd.

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc III YEAR MATHEMATICS SYLLABUS 2017-18 NUMERICAL ANALYSIS

SEMESTER-V PAPER-VI (A)Objectives:This course introduces students to numerical methods of solving differential equations and general interpolation.

UNIT-I:Errors in Numerical Computations: Numbers and their Accuracy, Errors and their Computation, Absolute, Relative and percentage errors, a general error formula. Solution of Algebraic and Transcendental Equations. The bisection method, The iteration method, The method of false position, Newton-Raphson method, Generalized Newton-Raphson method, order of convergence, Ramanujan’s method.

UNIT-II: Interpolation: Definition, Forward differences, Backward differences, Central Differences, Symbolic relations, Detection of errors by use of D.Tables. Differences of a polynomial. Newton’s forward and backward interpolation formulae , central difference interpolation, Gauss central difference formula,Stirling’s central difference formula..

UNIT-III:.Interpolation: Interpolation with unevenly spaced points, Lagrange’s formula, divided differences and their properties, Newton’s general interpolationNumerical Differentiation & Integration: Numerical differentiation, Numerical integration, Trapezoidal rule, Simpson’s 1/3 – rule, Simpson’s 3/8 – rule.

UNIT-IV:Numerical solution of ordinary differential equations : Introduction, Solution by Taylor’s Series, Picard’s method of successive approximations, Euler’s method, Modified Euler’s method, Runge – Kutta methods.Solution of linear systems- Iterative methods: Jacobi’s method, Gauss-siedal method,

Prescribed text Book: Scope as in Introductory Methods of Numerical Analysis by S.S. Sastry,

Prentice Hall India (4th Edition.), Chapter -1 (1. 2, 1. 3, 1.4,1.5), Chapter - 2 (2.2 – 2.6); chapter 3

(3.1, 3.3 to 3.7.3.9,3.9.1, 3.10,3.10.1,), Chapter.- 5(5.1,5.2,5.4 to 5.4.3.), Chapter - 7 (7.1 to 7.5);

chapter 8(8.3.1,8.3.2).

Reference Books:

1. Numerical Methods by B.S.Grewal

2. Finite Differences and Numerical Analysis by H.C. Saxena S. Chand and Company, New Del

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc III YEAR MATHEMATICS SYLLABUS 2017-18 SEMESTER-V PAPER-VI (B)

PAPER TITLE: LAPLACE AND INVERSE LAPLACE TRANSFORMSObjectives: This Course deals with Laplace transforms, Inverse Laplace transforms and it’s applications.

UNIT-I:

Definition of Laplace transform, linearity property piecewise continuous function, Existence of Laplace transform, Functions of exponential order and of class A, First and second shifting theorems of Laplace transform, change of scale property laplace transform of derivatives, final and initial value theorems, Laplace transform of integrals, multiplication by t, division by t, Laplace transform of periodic functions and error function, Beta function and Gamma functions.

UNIT-II: Definition of inverse Laplace transform linearity property, first and second shifting theorems of inverse Laplace transforms, change of scale property, division by p, convolution theorem, Heavisides expansion formula (with proofs and applications)

APPLICATIONS OF LAPLACE TRANSFORMS

UNIT-III: Solution of ordinary differential equations with constant coefficients, and variable coefficients, solution simultaneous ordinary differential equations.

UNIT-IV:Solution of partial differential equations, boundary value problems, heat equation, wave equation, Laplace equation and Integral equations.

Reference Books:

1. Integral transforms : Vasistha & Gupta.

(Krishna prakashan Mandir)

2. Fourier series and : Churchill

Boundary value problems (Tata Mc Graw- Hill publishing

Company Ltd.)

3. Higher Engineering : Dr. B.S.Grewal.

Mathematics

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc III YEAR MATHEMATICS SYLLABUS 2017-18 SEMESTER-VI PAPER-VII,LINEAR ALGEBRA & VECTOR CALCULUS

Linear AlgebraUNIT-I: The adjoint or transpose of a linear transformation, Sylvester’s law of nullity, rank of a matrix-echelon form,normal form, system of equations , characteristic values and characteristic vectors, Cayley- Hamilton theorem, Diagonalizable operators.

UNIT-II:Inner product spaces, Euclidean and unitary spaces, Norm or length of a vector, Schwartz

inequality, Orthogonality, Orthonormal set, complete orthonormal set, Gram - Schmidt orthogonalisation process.Prescribed text book:Linear Algebra by J.N.Sharma and A.R.Vasista, Krishna Prakasham Mandir, Meerut-250002. Reference Books: 1. Linear Algebra by Kenneth Hoffman and Ray Kunze, Pearson Education low priced edition), New Delhi2. Linear Algebra by Stephen H. Friedberg et al Prentice Hall of India Pvt. Ltd. 4th edition 2007.

Vector CalculusUNIT-III:Vector differentiation. Ordinary derivatives of vectors, Space curves, Continuity, Differentiability, Gradient, Divergence, Curl operators, Formulae involving these operators.

UNIT-IV:Vector integration, theorems of Greens, Gauss and Stokes theorem in plane and applications of these theorems.

Prescribed text Book: Vector Analysis by Murray. R.Spiegel, Schaum Series Publishing Company.Reference Books: 1.Text book of vector Analysis by Shanti Narayana and P. K. Mittal,S. Chand & Company Ltd, New Delhi.2. Mathematical Analysis by S.C. Mallik and Savitha Arora, Wiley Eastern Ltd.

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc III YEAR MATHEMATICS SYLLABUS 2017-18 SEMESTER-VI PAPER – VIII (a)

INTERGRAL TRANSFORMSFOURIER TRANSFORMS:UNIT-I:

Dirichlet’s conditions, Fourier integral formula (without proof), Fourier transform, Inverse Theorem for Fourier transform, Fourier sine and cosine transforms and their inversion formulae.

UNIT-II: Linearity property of Fourier transforms, change of scale property, shifting theorem, modulation theorem, and convolution theorem of Fourier transforms, Parseval’s identity.

FINITE FOURIER TRANSFORMS AND FOURIER SERIES:

UNIT-III: Finite Fourier sine transforms, Inversion formula for sine transforms, and finite Fourier cosine transforms, applications of Fourier transforms initial and boundary value problems.

UNIT-IV:

Fourier series in the interval (-l, l), (- , ), ( 0, 2), half-range sine and cosine series in the interval (0, l), and ( 0, ).

Reference Books:

1. Integral transforms : Vasistha & Gupta.

(Krishna prakashan Mandir)

2. Fourier series and : Churchill

Boundary value problems (Tata Mc Graw- Hill publishing

Company Ltd.)

3. Higher Engineering : Dr. B.S.Grewal.

Mathematics

OSMANIA UNIVERSITY COLLEGE FOR WOMEN, KOTI, HYDERABADB.Sc III YEAR MATHEMATICS SYLLABUS 2017-18

SEMESTER-VI PAPER-VIII (b)PAPER TITLE: NUMBER THEORY & CRYPTOGRAPHY

Objectives: This course aims to give the elementary ideas of Number theory which have number of applications in Cryptography.

Unit I: (Number theory) Introduction, GCD, LCM of integers, Division algorithm, Euler’s function, Prime Numbers, The fundamental theorem of arithmetic, Fermat’s theorem, Wilson’s Theorem, Distribution of primes, Fermat & Mersenne Primes, Primality Test.

Unit II: (Modular Arithmetic) Congruences, Linear congruences, Simultaneous linear congruences, Chinese remainder theorem, congruences mod prime power.Finite Fields, Finite fields of GF(P), Finite fields of GF(2n).

Unit III: (Introduction to cryptography) Some simple crypto systems, enciphering matrices.

Unit IV: The idea of public key cryptography , RSA Algorithm.

Prescribed Text books:Unit I & II scope as in “An Introduction to Theory of Numbers” by Niven and Zukerman, Willey Eastern Publications.Unit III & IV scope as in “A course in Number theory and cryptography” by Neal Koblitz, 2nd

edition, Springer publications.References: